A048942 a(n) is twice the coefficient of the radical part in the fundamental unit of Q(sqrt(A000037(n))) where A000037 lists the nonsquare numbers (Version 1).

2, 2, 1, 4, 6, 1, 2, 6, 1, 1, 8, 2, 2, 8, 78, 1, 1, 84, 10, 2, 2, 10, 3, 1, 4, 546, 1, 8, 12, 2, 2, 12, 8, 1, 10, 4, 1062, 3, 1, 7176, 14, 2, 2, 14, 5, 1, 132, 24, 4, 40, 26, 138, 1, 5, 16, 2, 2, 16, 11934, 1, 3, 60, 826, 4, 250, 10, 6, 39, 1, 12, 18, 2, 2, 18

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn

Comments

From Sean A. Irvine, Jul 16 2021: (Start)

These values are computed by Algorithm 5.7.2 in Cohen.

Other methods of computation (see A346420) give different results, with the first difference at n=14.

(End)

a(n) is the smallest positive integer y satisfying the Pell equation x^2 - D*y^2 = +-4, where D = A000037(n). - Jinyuan Wang, Sep 08 2021

References

• Henri Cohen, A Course in Computational Algebraic Number Theory, Springer-Verlag, 1993.

Links

• Jinyuan Wang, Table of n, a(n) for n = 1..1000
• S. R. Finch, Class number theory
• Steven R. Finch, Class number theory [Cached copy, with permission of the author]
• Sean A. Irvine, Java program (github)
• Eric Weisstein's World of Mathematics, Fundamental Unit.

Crossrefs

Cf. A000037, A007913, A048941, A346419, A346420.

Programs

PARI

a(n) = my(A, D=n+(1+sqrtint(4*n))\2, d=sqrtint(D), p, q, t, u1, u2, v1, v2); if(d%2==D%2, p=d, p=d-1); u1=-p; u2=2; v1=1; v2=0; q=2; while(v2==0 || q!=t, A=(p+d)\q; t=p; p=A*q-p; if(t==p && v2!=0, return(2*u2*v2/q), t=A*u2+u1; u1=u2; u2=t; t=A*v2+v1; v1=v2; v2=t; t=q; q=(D-p^2)/q)); (u1*v2+u2*v1)/q; \\ Jinyuan Wang, Sep 08 2021

Extensions

Name edited by Michel Marcus, Jun 26 2020

Entry revised by Sean A. Irvine, Jul 16 2021